1. Lectures 14-15 (Ch. 28) Sources of Magnetic Field 1. B of a moving charge 2. Bio-Savarat law 3. Superposition principle 4. Force between two currents 5. Flux of B 6. Amper’s Law 7. Bohr’s magneton 8. 4 types of magnetic materials

2. B of a moving charge

4. Compare electric and magnetic forces

5. Biot & Savart law ( B of a segment of a current)

7. B of a loop of a current

8. Electromagnet (magnetic coil or solenoid) N loops of the current) Most of magnets used in industry are electromagnets. Attraction is due to magnetization of iron items. (see the end of the previous lecture) For closely spaced loops (each loop at the same distance to observation point) of the same radius: In the center of the coil (x=0) N times stronger than from 1 loop

9. B of a stright wire Long wire: a>>x

10. Oersted’s experiments and Oersted’s RH rule Hans Christian Ørsted (Oersted) (1777 – 1851) 1820: current produces B

12. Superposition principle

13. Examples. Find B at point P.

14. A force between two currents NB: currents in the same direction attract each other. Currents in the opposite directions repeal each other. It’s different from charges of the same or opposite signs!

15. Example. Magnetic bottle again (see also lecture 12) Now we understand also the structure of magnetic field between as a result of superposition of two fields produced by two coils. NB: rotation direction of a positive charge is opposite to the current in the coils. Repulsive interaction of opposite currents results in trapping the charged particles in the bottle.

16. Example. Find the force acting on each piece of a rectangular conductor and the net force acting on the rectangular conductor.

17. Flux of B Gauss’s law for E: Gauss’s law for B: Magnetic lines are closed lines. There are no magnetic carges.

18. Circulation of B Example : Find a circulation of B produced by a current in the long straight wire for suggested integration passes and circulation directions.

19. Arbitrary shape of the closed line

20. Amper’s law Example. Find a circulation. Amper’s law allows one to find B for symmetric configurations of current.

21. Conducting cylinder

22. Coaxial cable Self-shielding

23. Example. An infinite current sheet. There are n conductors per unite length. Each of them carries a current, I. Find B.

24. Example. Prove that if in the absence of currents B is unidirectional it has to be uniform. B1B1 B2B2

25. Solenoid (N loops) L L’ In the long solenoid (L>>a) B at the exit =(1/2) B in the center

28. Toroidal solenoid (toroid)

29. Example. Find a) the net force on the loop and b)the flux of B through the front surface of the loop. I 2I a b d L

30. Bohr’s magneton Planck’s constant Bohr’s magneton e Different symmetries of e destributions S-state(L=0,μ L =0) 1e:S≠0,μ s ≠ 0 2e:Stotal=0, μ s =0 P-state:L= μ L ≠ 0

31. 4 types of magnetic materials 1.Paramagnetics:μ i ≠0, Under the action of B 0 alignment of μ i struggles with chaotic thermal motion, resulting in Curie’s law Inside the material : Magnetic susceptibility Relative magnetic permeability

32. 2. Ferromagnetics μ i ≠0, M ≠0 in domains

33. Electromagnets typically contain a ferromagnetic core In the absence of an iron core: In the presence of an iron core: Magnetic permeability Narrow hysteresis loop, Often use superconducting wires

34. Both paramagnetics and ferromagnetics are attracted to the magnet NS B 1. Alignment (randomly oriented μ or M become parallel to B, i.e. material becomes magnetized ) 2. Attraction(opposite poles attract each other) SN B SN

35. NS B0B0 1. Induced M is anti- parallel to B 0 2. Repulsion(similar poles repel each other) SN B0B0 NS Diamagnetics In the absence of magnetic field μ i =0,M=0 In the presence of magnetic field M is induced in the direction opposite to external magnetic field (consequence of Lens’s low, see next lecture) These materials a repelled by magnets, though this repulsion is very week.

37. Superconductors (R=0) Perfect diamagnetics: K m =0, No magnetic field inside of superconductors Meisner’s effect